Euler's Homogeneous Function Theorem

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Let $\map f{x,y}$ be a homogeneous function of order $n$ so that

$\map f{tx,ty}=t^n\map f{x,y}$

Then:

$x\dfrac{\partial f}{\partial x}+y\dfrac{\partial f}{\partial y}=n\map f{x,y}$

Proof

Define $x'=xt$ and $y'=yt$. By chain rule

\(\ds nt^{n-1}\map f{x,y}\)

\(=\)

\(\ds \frac{\partial f}{\partial x'}\frac{\partial x'}{\partial t}+\frac{\partial f}{\partial y'}\frac{\partial y'}{\partial t}\)

\(\ds \)

\(=\)

\(\ds x\frac{\partial f}{\partial x'}+y\frac{\partial f}{\partial y'}\)

\(\ds \)

\(=\)

\(\ds x\frac{\partial f}{\partial xt}+y\frac{\partial f}{\partial yt}.\)

Let $t=1$, then

$x\dfrac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=n\map f{x,y}. $

Generalization

This can be generalized to an arbitrary number of variables

$\ds\sum_i x_i\frac{\partial f}{\partial x_i}=n\map f {\ldots,x_i,\ldots} $

Source of Name

This entry was named for Leonhard Paul Euler.

Source

• Weisstein, Eric W. "Euler's Homogeneous Function Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EulersHomogeneousFunctionTheorem.html